Search results for "Lindelöf space"
showing 3 items of 3 documents
Cardinal invariants of cellular Lindelof spaces
2018
A space X is said to be cellular-Lindelof if for every cellular family $$\mathcal {U}$$ there is a Lindelof subspace L of X which meets every element of $$\mathcal {U}$$ . Cellular-Lindelof spaces generalize both Lindelof spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelof monotonically normal space is Lindelof and that every cellular-Lindelof space with a regular $$G_\delta $$ -diagonal has cardinality at most $$2^\mathfrak {c}$$ . We also prove that every normal cellular-Lindelof first-countable space has cardinality at most continuum under $$2^{<\mathfrak {c}}=\mathfrak {c}$$ and that every normal cellular-Lindel…
P-spaces and the Whyburn property
2009
We investigate the Whyburn and weakly Whyburn property in the class of $P$-spaces, that is spaces where every countable intersection of open sets is open. We construct examples of non-weakly Whyburn $P$-spaces of size continuum, thus giving a negative answer under CH to a question of Pelant, Tkachenko, Tkachuk and Wilson. In addition, we show that the weak Kurepa Hypothesis (a set-theoretic assumption weaker than CH) implies the existence of a non-weakly Whyburn $P$-space of size $\aleph_2$. Finally, we consider the behavior of the above-mentioned properties under products; we show in particular that the product of a Lindel\"of weakly Whyburn P-space and a Lindel\"of Whyburn $P$-space is we…
On the cardinality of almost discretely Lindelof spaces
2016
A space is said to be almost discretely Lindelof if every discrete subset can be covered by a Lindelof subspace. Juhasz et al. (Weakly linearly Lindelof monotonically normal spaces are Lindelof, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelof first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhasz et al. (First-countable and almost discretely Lindelof $$T_3$$ spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few rel…